On the Minimum Distance of Turbo Codes With Quadratic Permutation Polynomial Interleavers

An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementations. Also, the recently proposed quadratic permutation polynomial (QPP)-based interleavers by Sun and Takeshita have provided excellent performance for short-to-medium block lengths, and have been selected for the 3GPP LTE standard. In this paper, we derive some upper bounds on the best achievable minimum distance d(min) of QPP-based conventional binary turbo codes (with tailbiting termination, or dual termination when the interleaver length N is sufficiently large) that are tight for larger block sizes. In particular, we show that the minimum distance is at most 2(2(v+1) +9 independent of the interleaver length, when there exists an inverse polynomial of degree two, where v is the degree of the primitive feedback and monic feedforward polynomials. However, allowing the QPP to have no inverse polynomials of degree two may give strictly larger minimum distances (and lower multiplicities). In particular, we provide several QPPs with no quadratic inverse for some of the 3GPP LTE interleaver lengths giving a d(min) with the 3GPP LTE constituent encoders which is strictly larger than 50. For instance, we have found a QPP for N = 6016 which gives an estimated d(min) of 57. Furthermore, we provide the exact minimum distances and the corresponding multiplicities for all 3GPP LTE turbo codes (with dual termination) which shows that the best minimum distance is 51. Finally, we compute the best achievable minimum distances with QPP interleavers for all 3GPP LTE interleaver lengths N <= 4096, and compare these minimum distances with the ones we get when using the 3GPP LTE polynomials.